File size: 4,041 Bytes
dd14c26 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 | import OAI.Probability.MatroidProphet.Reverse.BirthSafety
namespace OAI
namespace MatroidProphet
open Finset
variable {α : Type*} [Fintype α] [DecidableEq α]
attribute [local instance] Classical.propDecidable
lemma reverseCost_add_steps (M : Matroid α) (hE : M.E = Set.univ)
(κ : ℕ) (D : ℕ → Set α) (G : ℕ → Finset α) (n : ℕ)
(cost : ℤ → (ℕ → Set α) → (ℕ → Set α) → ℝ)
(m r : ℕ) (k : ℤ) (S : ℕ → Set α) (C : Finset α) :
reverseCost M hE κ D G n cost (m+r) k S C =
reverseCost M hE κ D G n cost m k S C +
reverseCost M hE κ D G n cost r (k-m) (reverseFinal M hE κ D G n m k S C) C := by
induction m generalizing k S with
| zero => simp only [reverseCost, reverseFinal, Nat.cast_zero, sub_zero, zero_add]
| succ m ih =>
rw [Nat.succ_add, reverseCost, reverseCost, reverseFinal, ih]
have ht : k - 1 - (m:ℤ) = k - ((m+1:ℕ):ℤ) := by omega
rw [ht]
ring
lemma reverseCost_eq_sum (M : Matroid α) (hE : M.E = Set.univ)
(κ : ℕ) (D : ℕ → Set α) (G : ℕ → Finset α) (n : ℕ)
(cost : ℤ → (ℕ → Set α) → (ℕ → Set α) → ℝ)
(m : ℕ) (k : ℤ) (S : ℕ → Set α) (C : Finset α) :
reverseCost M hE κ D G n cost m k S C =
∑ i ∈ range m, cost (k-i) (reverseFinal M hE κ D G n i k S C)
((reverseStepTree M hE κ (k-i) D (reverseFinal M hE κ D G n i k S C) G n).run C) := by
induction m with
| zero => simp only [reverseCost, range_zero, sum_empty]
| succ m ih =>
rw [reverseCost_add_steps M hE κ D G n cost m 1, ih, sum_range_succ]
simp only [reverseCost, add_zero]
noncomputable def pathExitSquare (M : Matroid α) (hE : M.E = Set.univ)
(κ : ℕ) (D : ℕ → Set α) (G : ℕ → Finset α) (q : α → ℝ) (d : α) (h : ℕ) (m : ℕ)
(C : Finset α) : ℝ :=
if d ∈ nominalPath M hE κ D (fun i => (C : Set α) ∩ (G i : Set α)) h (-(m:ℤ)) then
(reverseHazard M hE κ D G q d h (-(m:ℤ)-1)
(fun j => nominalPath M hE κ D (fun i => (C : Set α) ∩ (G i : Set α)) j (-(m:ℤ))))^2 else 0
lemma reverse_square_eq_path_sum (M : Matroid α) (hE : M.E = Set.univ)
(κ : ℕ) (D : ℕ → Set α) (G : ℕ → Finset α) (n : ℕ)
(hG : Pairwise (fun i j => Disjoint (G i) (G j))) (q : α → ℝ)
(d : α) (h : ℕ) (hh : h ≤ n) (m : ℕ) :
reverseMeanCost M hE κ D G n q (squareHazardCost M hE κ D G q d h) m (-1) (fun _ => Set.univ) =
∑ i ∈ range m, bitsExpectation q univ (pathExitSquare M hE κ D G q d h i) := by
unfold reverseMeanCost
rw [← bitsExpectation_sum]
apply bitsExpectation_congr
intro C hC
rw [reverseCost_eq_sum]
apply sum_congr rfl
intro i hi
have hS : ∀ j ≤ n, (fun _ => Set.univ) j =
nominalPath M hE κ D (fun l => (C : Set α) ∩ (G l : Set α)) j (-1+1) := by
intro j hj
exact (nominalPath_nonnegative M hE κ D _ j (by omega)).symm
have hf (j : ℕ) (hj : j ≤ n) : reverseFinal M hE κ D G n i (-1) (fun _ => Set.univ) C j =
nominalPath M hE κ D (fun l => (C : Set α) ∩ (G l : Set α)) j (-(i:ℤ)) := by
simpa only [neg_add_cancel, zero_sub] using reverseFinal_path M hE κ D G n hG i (-1) _ C hS hj
have ht : (-1:ℤ) - i = -(i:ℤ)-1 := by omega
simp only [squareHazardCost, pathExitSquare, ht, hf h hh]
rw [reverseHazard_state_congr M hE κ D _ _ G q _ h d (fun j hj => hf j (by omega))]
theorem path_squared_exit_mass (M : Matroid α) (hE : M.E = Set.univ)
(κ : ℕ) (D : ℕ → Set α) (G : ℕ → Finset α)
(hG : Pairwise (fun i j => Disjoint (G i) (G j))) (q : α → ℝ)
(hq0 : ∀ e, 0 ≤ q e) (hq1 : ∀ e, q e ≤ 1)
(d : α) (hd : d ∉ M.closure ∅) (h : ℕ) :
(((2:ℝ)^12)⁻¹) / 2 ≤
∑ i ∈ range (pathHorizon h+1), bitsExpectation q univ (pathExitSquare M hE κ D G q d h i) := by
rw [← reverse_square_eq_path_sum M hE κ D G h hG q d h le_rfl]
exact reverse_squared_exit_mass M hE κ D G h hG q hq0 hq1 d hd h le_rfl
end MatroidProphet
end OAI
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